This dissertation is composed of three chapters that develop new estimation methods for several models of panel data. The first and third chapters are mainly concerned with understanding and aproximating the structure of optimal instruments for estimating dynamic panel data models with cross-sectional dependence in the case of the first chapter, and non-linear panel data models with strictly exogeneous covariates in the case of the third chapter. The second chapter is concerned with additional restrictions that can be used to estimate non-linear dynamic panel data models. The first chapter considers the estimation of dynamic panel data models when data are suspected to exhibit cross-sectional dependence. A new estimator is defined that uses cross-sectional dependence for efficiency while being robust to the misspecification of the form of the cross-sectional dependence. I show that using cross-sectional dependence for estimation is important to obtain an estimator that is more accurate than existing estimators. This new estimator also uses nuisance parameters parsimoniously so that it exhibits good small sample properties even when the number of available moment conditions is large. As an empirical application, I estimate the effect of attending private school on student achievement using a value added model.The second chapter considers the instrumental variable estimation of non-linear models of panel data with multiplicative unobserved effects where instrumental variables are predetermined as opposed to strictly exogenous. Existing estimators for these models suffer from a weak instrumental variable problem, which can cause them to be too inaccurate to be reliable. In this chapter I present additional sets of restrictions that can be used for more precise estimation. Monte Carlo simulations show that using these additional moment conditions improves the precision of the estimators significantly and hence should facilitate the use of these models.In the third chapter I study the efficiency of the Poisson Fixed Effects estimator. The Poisson fixed effects estimator is a conditional maximum likelihood estimator and as such is consistent under specific distributional assumptions. It has also been shown to be consistent under significantly weaker restrictions on the conditional mean function only. I show that the Poisson Fixed Effects estimator is asymptotically efficient in the class of estimators that are consistent under restrictions on the conditional mean function, as long as the assumptions of equal conditional mean and variance and zero conditional serial correlation are satisfied. I then define another estimator that is optimal under more general conditions. I use Monte Carlo simulations to investigate the small-sample performance of this new estimator compared to the Poisson fixed effects estimator.
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